An extended cell-based smoothed discrete shear gap method (XCS-FEM-DSG3) for free vibration analysis of cracked Reissner-Mindlin shells
M. H. NGUYEN-THOI1,2,L. Le-ANH1,2,V. Ho-HUU1,2,H. Dang-TRUNG1,2,T. NGUYEN-THOI1,2,*()
1. Division of Computational Mathematics and Engineering (CME), Institute for Computational Science (INCOS), Ton Duc Thang University, Hochiminh city, Vietnam 2. Faculty of Civil Engineering, Ton Duc Thang University, Hochiminh city, Vietnam
A cell-based smoothed discrete shear gap method (CS-FEM-DSG3) was recently proposed and proven to be robust for free vibration analyses of Reissner-Mindlin shell. The method improves significantly the accuracy of the solution due to softening effect of the cell-based strain smoothing technique. In addition, due to using only three-node triangular elements generated automatically, the CS-FEM-DSG3 can be applied flexibly for arbitrary complicated geometric domains. However so far, the CS-FEM-DSG3 has been only developed for analyzing intact structures without possessing internal cracks. The paper hence tries to extend the CS-FEM-DSG3 for free vibration analysis of cracked Reissner-Mindlin shells by integrating the original CS-FEM-DSG3 with discontinuous and crack−tip singular enrichment functions of the extended finite element method (XFEM) to give a so-called extended cell-based smoothed discrete shear gap method (XCS-FEM-DSG3). The accuracy and reliability of the novel XCS-FEM-DSG3 for free vibration analysis of cracked Reissner-Mindlin shells are investigated through solving three numerical examples and comparing with commercial software ANSYS.
Kwon Y W. Development of finite element shape functions with derivative singularity. Computers & Structures, 1988, 30(5): 1159–1163
2
Krawczuk M. Rectangular shell finite element with an open crack. Finite Elements in Analysis and Design, 1994, 15(3): 233–253
3
Liu R, Zhang T, Wu X, Wang C. Determination of stress intensity factors for a cracked shell under bending with improved shell theories. Journal of Aerospace Engineering, 2006, 19(1): 21–28
4
Vaziri A, Estekanchi H E. Buckling of cracked cylindrical thin shells under combined internal pressure and axial compression. Thin-walled Structures, 2006, 44(2): 141–151
5
Fu J, To C W S. Bulging factors and geometrically nonlinear responses of cracked shell structures under internal pressure. Engineering Structures, 2012, 41: 456–463
6
Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering, 1999, 45(5): 601–620
7
Moës N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering, 1999, 46: 131–150
8
Stolarska M, Chopp D L, Moës N, Belytschko T. Modelling crack growth by level sets in the extended finite element method. International Journal for Numerical Methods in Engineering, 2001, 51(8): 943–960
9
Bachene M, Tiberkak R, Rechak S. Vibration analysis of cracked plates using the extended finite element method. Archive of Applied Mechanics, 2009, 79(3): 249–262
10
Natarajan S, Baiz P M, Bordas S, Rabczuk T, Kerfriden P. Natural frequencies of cracked functionally graded material plates by the extended finite element method. Composite Structures, 2011, 93(11): 3082–3092
11
Rabczuk T, Areias P M A. A meshfree thin shell for arbitrary evolving cracks based on an external enrichment. CMES-Computer Modeling in Engineering and Sciences, 2006, 16: 115–130
12
Rabczuk T, Areias P M A, Belytschko T. A meshfree thin shell method for non-linear dynamic fracture. International Journal for Numerical Methods in Engineering, 2007, 72(5): 524–548
13
Zhuang X, Augarde C E, Mathisen K M. Fracture modeling using meshless methods and level sets in 3D: Framework and modeling. International Journal for Numerical Methods in Engineering, 2012, 92(11): 969–998
14
Chau-Dinh T, Zi G, Lee P S, Rabczuk T, Song J H. Phantom-node method for shell models with arbitrary cracks. Computers & Structures, 2012, 92−93: 242–256
15
Ghorashi S S, Valizadeh N, Mohammadi S, Rabczuk T. T-spline based XIGA for fracture analysis of orthotropic media. Computers & Structures, 2015, 147: 138–146
16
Nguyen-Thanh N, Valizadeh N, Nguyen M N, Nguyen-Xuan H, Zhuang X, Areias P, Zi G, Bazilevs Y, De Lorenzis L, Rabczuk T. An extended isogeometric thin shell analysis based on Kirchhoff−Love theory. Computer Methods in Applied Mechanics and Engineering, 2015, 284: 265–291
17
Areias P, Rabczuk T. Finite strain fracture of plates and shells with configurational forces and edge rotations. International Journal for Numerical Methods in Engineering, 2013, 94(12): 1099–1122
https://doi.org/10.1002/nme.4477
18
Areias P, Rabczuk T, Camanho P P. Initially rigid cohesive laws and fracture based on edge rotations. Computational Mechanics, 2013, 52(4): 931–947
19
Areias P, Rabczuk T, Dias-da-Costa D. Element-wise fracture algorithm based on rotation of edges. Engineering Fracture Mechanics, 2013, 110: 113–137
20
Areias P, Rabczuk T, Camanho P P. Finite strain fracture of 2D problems with injected anisotropic softening elements. Theoretical and Applied Fracture Mechanics, 2014, 72: 50–63
21
Liu GR, Nguyen-Thoi T. Smoothed Finite Element Methods. NewYork: Taylor and Francis Group, 2010
22
Liu G R, Nguyen-Thoi T, Nguyen-Xuan H, Dai K Y, Lam K Y. On the essence and the evaluation of the shape functions for the smoothed finite element method (SFEM). International Journal for Numerical Methods in Engineering, 2009, 77(13): 1863–1869
23
Nguyen T T, Liu G R, Dai K Y, Lam K Y. Selective smoothed finite element method. Tsinghua Science and Technology, 2007, 12(5): 497–508
24
Liu G R, Dai K Y, Nguyen T T. A smoothed finite element method for mechanics problems. Computational Mechanics, 2007, 39(6): 859–877
25
Nguyen-Thoi T, Liu G R, Nguyen-Xuan H. Additional properties of the node-based smoothed finite element method (NS-FEM) for solid mechanics problems. International Journal of Computational Methods, 2009, 06(04): 633–666
26
Nguyen-Thoi T, Liu G R, Nguyen-Xuan H, Nguyen-Tran C. Adaptive analysis using the node-based smoothed finite element method (NS-FEM). International Journal for Numerical Methods in Biomedical Engineering, 2011, 27(2): 198–218
27
Liu G R, Nguyen-Thoi T, Nguyen-Xuan H, Lam K Y. A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems. Computers & Structures, 2009, 87(1−2): 14–26
28
Nguyen-Thoi T, Liu G R, Nguyen-Xuan H, Nguyen-Tran C. An n-sided polygonal edge-based smoothed finite element method (nES-FEM) for solid mechanics. International Journal for Numerical Methods in Biomedical Engineering, 2011, 27: 1446–1472
29
Liu G R, Nguyen-Thoi T, Lam K Y. An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses of solids. Journal of Sound and Vibration, 2009, 320(4−5): 1100–1130
30
Nguyen-Thoi T, Liu G R, Lam K Y, Zhang G Y. A face-based smoothed finite element method (FS-FEM) for 3D linear and geometrically non-linear solid mechanics problems using 4-node tetrahedral elements. International Journal for Numerical Methods in Engineering, 2009, 78(3): 324–353
31
Liu G R, Nguyen-Xuan H, Nguyen-Thoi T, Xu X. A novel Galerkin-like weakform and a superconvergent alpha finite element method (S-alpha FEM) for mechanics problems using triangular meshes. Journal of Computational Physics, 2009, 228(11): 4055–4087
32
Liu G R, Nguyen-Thoi T, Lam K Y. A novel alpha finite element method (αFEM) for exact solution to mechanics problems using triangular and tetrahedral elements. Computer Methods in Applied Mechanics and Engineering, 2008, 197(45−48): 3883–3897
33
Liu G R, Nguyen-Xuan H, Nguyen-Thoi T. A variationally consistent alpha FEM (VC alpha FEM) for solution bounds and nearly exact solution to solid mechanics problems using quadrilateral elements. International Journal for Numerical Methods in Engineering, 2011, 85(4): 461–497
34
Liu G R, Nguyen-Thoi T, Lam K Y. A novel FEM by scaling the gradient of strains with factor alpha (alpha FEM). Computational Mechanics, 2009, 43(3): 369–391
35
Liu G R, Nguyen T T, Dai K Y, Lam K Y. Theoretical aspects of the smoothed finite element method (SFEM). International Journal for Numerical Methods in Engineering, 2007, 71(8): 902–930
36
Liu G R, Nguyen-Xuan H, Nguyen-Thoi T. A theoretical study on the smoothed FEM (S-FEM) models: Properties, accuracy and convergence rates. International Journal for Numerical Methods in Engineering, 2010, 84(10): 1222–1256
37
Nguyen-Xuan H, Rabczuk T, Bordas S, Debongnie J F. A smoothed finite element method for plate analysis. Computer Methods in Applied Mechanics and Engineering, 2008, 197(13−16): 1184–1203
38
Nguyen-Xuan H, Liu G R, Thai-Hoang C, Nguyen-Thoi T. An edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysis of Reissner−Mindlin plates. Computer Methods in Applied Mechanics and Engineering, 2010, 199(9−12): 471–489
39
Nguyen-Xuan H, Rabczuk T, Nguyen-Thanh N, Nguyen-Thoi T, Bordas S. A node-based smoothed finite element method with stabilized discrete shear gap technique for analysis of Reissner−Mindlin plates. Computational Mechanics, 2010, 46(5): 679–701
40
Nguyen-Xuan H, Tran L V, Nguyen-Thoi T, Vu-Do H C. Analysis of functionally graded plates using an edge-based smoothed finite element method. Composite Structures, 2011, 93(11): 3019–3039
41
Nguyen-Xuan H, Tran L V, Thai C H, Nguyen-Thoi T. Analysis of functionally graded plates by an efficient finite element method with node-based strain smoothing. Thin-walled Structures, 2012, 54: 1–18
42
Thai C H, Tran L V, Tran D T, Nguyen-Thoi T, Nguyen-Xuan H. Analysis of laminated composite plates using higher-order shear deformation plate theory and node-based smoothed discrete shear gap method. Applied Mathematical Modelling, 2012, 36(11): 5657–5677
43
Nguyen-Thoi T, Bui-Xuan T, Phung-Van P, Nguyen-Xuan H, Ngo-Thanh P. Static, free vibration and buckling analyses of stiffened plates by CS-FEM-DSG3 using triangular elements. Computers & Structures, 2013, 125: 100–113
44
Nguyen-Thoi T, Phung-Van P, Luong-Van H, Nguyen-Van H, Nguyen-Xuan H. A cell-based smoothed three-node Mindlin plate element (CS-MIN3) for static and free vibration analyses of plates. Computational Mechanics, 2013, 51(1): 65–81
45
Nguyen-Thoi T, Phung-Van P, Thai-Hoang C, Nguyen-Xuan H. A cell-based smoothed discrete shear gap method (CS-DSG3) using triangular elements for static and free vibration analyses of shell structures. International Journal of Mechanical Sciences, 2013, 74: 32–45
46
Phan-Dao H H, Nguyen-Xuan H, Thai-Hoang C, Nguyen-Thoi T, Rabczuk T. An edge-based smoothed finite element method for analysis of laminated composite plates. International Journal of Computational Methods, 2013, 10(01): 1340005
47
Phung-Van P, Nguyen-Thoi T, Tran L V, Nguyen-Xuan H. A cell-based smoothed discrete shear gap method (CS-DSG3) based on the C0-type higher-order shear deformation theory for static and free vibration analyses of functionally graded plates. Computational Materials Science, 2013, 79: 857–872
48
Luong-Van H, Nguyen-Thoi T, Liu G R, Phung-Van P. A cell-based smoothed finite element method using three-node shear-locking free Mindlin plate element (CS-FEM-MIN3) for dynamic response of laminated composite plates on viscoelastic foundation. Engineering Analysis with Boundary Elements, 2014, 42: 8–19
49
Nguyen-Thoi T, Bui-Xuan T, Phung-Van P, Nguyen-Hoang S, Nguyen-Xuan H. An edge-based smoothed three-node mindlin plate element (ES-MIN3) for static and free vibration analyses of plates. KSCE Journal of Civil Engineering, 2014, 18(4): 1072–1082
50
Phung-Van P, Nguyen-Thoi T, Luong-Van H, Lieu-Xuan Q. Geometrically nonlinear analysis of functionally graded plates using a cell-based smoothed three-node plate element (CS-MIN3) based on the C0-HSDT. Computer Methods in Applied Mechanics and Engineering, 2014, 270: 15–36
51
Phung-Van P, Nguyen-Thoi T, Le-Dinh T, Nguyen-Xuan H. Static and free vibration analyses and dynamic control of composite plates integrated with piezoelectric sensors and actuators by the cell-based smoothed discrete shear gap method (CS-FEM-DSG3). Smart Materials and Structures, 2013, 22(9): 17
52
Nguyen-Xuan H, Liu G R, Nguyen-Thoi T, Nguyen-Tran C. An edge-based smoothed finite element method for analysis of two-dimensional piezoelectric structures. Smart Materials and Structures, 2009, 18(6): 1–12
53
Liu G R, Chen L, Nguyen-Thoi T, Zeng K Y, Zhang G Y. A novel singular node-based smoothed finite element method (NS-FEM) for upper bound solutions of fracture problems. International Journal for Numerical Methods in Engineering, 2010, 83(11): 1466–1497
54
Nguyen-Thoi T, Liu G R, Vu-Do H C, Nguyen-Xuan H. A face-based smoothed finite element method (FS-FEM) for visco-elastoplastic analyses of 3D solids using tetrahedral mesh. Computer Methods in Applied Mechanics and Engineering, 2009, 198(41−44): 3479–3498
55
Nguyen-Thoi T, Vu-Do H C, Rabczuk T, Nguyen-Xuan H. A node-based smoothed finite element method (NS-FEM) for upper bound solution to visco-elastoplastic analyses of solids using triangular and tetrahedral meshes. Computer Methods in Applied Mechanics and Engineering, 2010, 199(45−48): 3005–3027
56
Nguyen-Thoi T, Liu G R, Vu-Do H C, Nguyen-Xuan H. An edge-based smoothed finite element method for visco-elastoplastic analyses of 2D solids using triangular mesh. Computational Mechanics, 2009, 45(1): 23–44
57
Nguyen-Xuan H, Rabczuk T, Nguyen-Thoi T, Tran T N, Nguyen-Thanh N. Computation of limit and shakedown loads using a node-based smoothed finite element method. International Journal for Numerical Methods in Engineering, 2012, 90(3): 287–310
58
Tran T N, Liu G R, Nguyen-Xuan H, Nguyen-Thoi T. An edge-based smoothed finite element method for primal−dual shakedown analysis of structures. International Journal for Numerical Methods in Engineering, 2010, 82: 917–938
59
Nguyen-Thoi T, Phung-Van P, Rabczuk T, Nguyen-Xuan H, Le-Van C. An application of the ES-FEM in solid domain for dynamic analysis of 2d fluid-solid interaction problems. International Journal of Computational Methods, 2013, 10
60
Nguyen-Thoi T, Phung-Van P, Rabczuk T, Nguyen-Xuan H, Le-Van C. Free and forced vibration analysis using the n-sided polygonal Cell-Based Smoothed Finite Element Method (NCS-FEM). International Journal of Computational Methods, 2013, 10(01): 1340008
61
Bletzinger K U, Bischoff M, Ramm E. A unified approach for shear-locking-free triangular and rectangular shell finite elements. Computers & Structures, 2000, 75(3): 321–334
62
Phung-Van P, Nguyen-Thoi T, Tran L V, Nguyen-Xuan H. A cell-based smoothed discrete shear gap method (CS-DSG3) based on the C-0-type higher-order shear deformation theory for static and free vibration analyses of functionally graded plates. Computational Materials Science, 2013, 79: 857–872
63
Phung-Van P, Nguyen-Thoi T, Dang-Trung H, Nguyen-Minh N. A cell-based smoothed discrete shear gap method (CS-FEM-DSG3) using layerwise theory based on the C0-type higher-order shear deformation for static and free vibration analyses of sandwich and composite plates. Composite Structures, 2014, 111: 553–565
64
Phung-Van P, Luong-Van H, Nguyen-Thoi T, Nguyen-Xuan H. A cell-based smoothed discrete shear gap method (CS-DSG3) based on the higher-order shear deformation theory for dynamic responses of Mindlin plates on the viscoelastic foundation subjected to a moving sprung vehicle. International Journal for Numerical Methods in Engineering, 2014, 98(13): 988–1014
65
Phung-Van P, Nguyen-Thoi T, Luong-Van H, Thai-Hoang C, Nguyen-Xuan H. A cell-based smoothed discrete shear gap method (CS-FEM-DSG3) using layerwise deformation theory for dynamic response of composite plates resting on viscoelastic foundation. Computer Methods in Applied Mechanics and Engineering, 2014, 272: 138–159
66
Bischoff M, Bletzinger K U. Stabilized DSG plate and shell elements. Trends in Computational structural mechanics. CIMNE. Barcelona, Spain, 2001
67
Lyly M, Stenberg R, Vihinen T. A stable bilinear element for the Reissner-Mindlin plate model. Computer Methods in Applied Mechanics and Engineering, 1993, 110(3−4): 343–357
68
Babuška I, Caloz G, Osborn J. Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM Journal on Numerical Analysis, 1994, 31(4): 945–981
69
Melenk J M. On Generalized Finite Element M<?Pub Caret?>ethods: University of Maryland, 1995
70
Babuška I, Melenk J. The partition of unity finite element method. International Journal for Numerical Methods in Engineering, 1997, 40(4): 727–758
71
Simone A, Duarte C A, Van der Giessen E. A Generalized Finite Element Method for polycrystals with discontinuous grain boundaries. International Journal for Numerical Methods in Engineering, 2006, 67(8): 1122–1145
72
Babuška I, Nistor V, Tarfulea N. Generalized finite element method for second-order elliptic operators with Dirichlet boundary conditions. Journal of Computational and Applied Mathematics, 2008, 218(1): 175–183
73
Dolbow J, Moës N, Belytschko T. Modeling fracture in Mindlin−Reissner plates with the extended finite element method. International Journal of Solids and Structures, 2000, 37(48−50): 7161–7183
74
Ventura G. On the elimination of quadrature subcells for discontinuous functions in the eXtended Finite-Element Method. International Journal for Numerical Methods in Engineering, 2006, 66(5): 761–795
